Lifshits Tails Caused by Anisotropic Decay: The Emergence of a Quantum-Classical Regime

We investigate Lifshits-tail behaviour of the integrated density of states for a wide class of Schrödinger operators with positive random potentials. The setting includes alloy-type and Poissonian random potentials. The considered (single-site) impurity potentials f: ?^d→[0,∞[ decay at infinity in an anisotropic way, for example, (f(x_{1},x_{2})sim (|x_{1}|^{alpha_{1}}+|x_{2}|^{alpha_{2}})^{-1}) as |(x₁,x₂)|→∞. As is expected from the isotropic situation, there is a so-called quantum regime with Lifshits exponent d/2 if both α₁ and α₂ are big enough, and there is a so-called classical regime with Lifshits exponent depending on α₁ and α₂ if both are small. In addition to this we find two new regimes where the Lifshits exponent exhibits a mixture of quantum and classical behaviour. Moreover, the transition lines between these regimes depend in a nontrivial way on α₁ and α₂ simultaneously.